A solvable model for octupole phonons

TL;DR

This paper introduces an exactly solvable shell-model based framework for describing octupole phonons in nuclei, providing explicit formulas for their energies and wave functions, applicable across different nuclear shell closures.

Contribution

It presents a new solvable model for octupole phonons that yields explicit expressions for their properties, bridging light and heavy nuclei cases.

Findings

Analytic wave functions proportional to Wigner 3j coefficients.

Exact validity in light nuclei with LS shell closures.

Good approximation in heavier nuclei with jj shell closures.

Abstract

A solvable model is proposed for the description of octupole phonons in closed-shell nuclei, formulated in terms of shell-model par\-ticle--hole excitations. With some simple assumptions concerning single-particle energies and two-body interactions, closed expressions are derived for the energy and wave function of the octupole phonon. In particular, it is shown that the components of the octupole phonon are proportional to Wigner $3j$ coefficients. This analytic wave function is proven to be exactly valid in light nuclei, which have $LS$ shell closures that coincide with those of the three-dimensional harmonic oscillator, and to be valid to a good approximation in heavier nuclei, which have $jj$ shell closures due to the spin--orbit interaction. The properties of the solvable model are compared with the results of a realistic shell-model calculation for $^{208}$Pb.